GAAG seminar - The arc-Floer conjecture: from arithmetic to symplectic topology

In the same philosophy as the Weil conjectures, Igusa's classical monodromy conjecture suggests a mysterious connection between the arithmetic and the topology of hypersurface singularities. Although it has been verified in several cases, the conjecture is still widely open, and the attempts to solve it have motivated new developments in different directions: motivic, nonarchimedean, model-theoretic, etc. In 2022, Budur, F. de Bobadilla, Lê and Nguyen suggested a new approach by exploding the symplectic properties of the Milnor fibration.

More precisely, the authors suggested the so-called arc-Floer conjecture, which states that the cohomology of contact loci coincides (up to a shift) with the Floer homology of the iterates of a symplectic monodromy. However, the authors formulated the conjecture based on analogies but not on actual evidence. In 2023, de la Bodega and de Lorenzo Poza verified it for plane curve singularities, providing the first piece of evidence supporting the arc-Floer conjecture. In this talk, we will give an overview of the problem and explain the proof for plane curves.